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Pauli matrices
(1900–1958), ca. 1924. Pauli received the in 1945, nominated by , for the .}} In and , the Pauli matrices are a set of three which are and . Usually indicated by the letter ( ), they are occasionally denoted by ( ) when used in connection with symmetries. They are : \begin{align} \sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \,. \end{align} These matrices are named after the physicist . In , they occur in the which takes into account the interaction of the of a particle with an external . Each Pauli matrix is , and together with the identity matrix (sometimes considered as the zeroth Pauli matrix ), the Pauli matrices form a for the real of Hermitian matrices. This means that any can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. Hermitian operators represent s in quantum mechanics, so the Pauli matrices span the space of observables of the -dimensional complex . In the context of Pauli's work, represents the observable corresponding to spin along the th coordinate axis in three-dimensional . The Pauli matrices (after multiplication by to make them ) also generate transformations in the sense of s: the matrices form a basis for the real Lie algebra \mathfrak{su}(2) , which to the special unitary group . The generated by the three matrices is to the of , and the algebra generated by is isomorphic to the . Algebraic properties All three of the Pauli matrices can be compacted into a single expression: : \sigma_a = \begin{pmatrix} \delta_{a3} & \delta_{a1} - i\delta_{a2}\\ \delta_{a1} + i\delta_{a2} & -\delta_{a3} \end{pmatrix} where }} is the , and is the , which equals +1 if b''}} and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of 1, 2, 3}}, in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations. The matrices are : : \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I where is the . The s and s of the Pauli matrices are: : \begin{align} \det \sigma_i &= -1, \\ \operatorname{tr} \sigma_i &= 0 . \end{align} From which, we can deduce that the of each are . With the inclusion of the identity matrix, (sometimes denoted ), the Pauli matrices form an orthogonal basis (in the sense of ) of the real of complex Hermitian matrices, \mathcal{H}_2(\mathbb{C}) , and the complex Hilbert space of all matrices, \mathcal{M}_{2,2}(\mathbb{C}) . Eigenvectors and eigenvalues Each of the ( ) Pauli matrices has two , and . Using a convention in which prior to normalization, the 1 is placed into the top and bottom positions of the + and – wavefunctions respectively, the corresponding are: : \begin{align} \psi_{x+} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, & \psi_{x-} &= \frac{1}{\sqrt{2}} \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \\ \psi_{y+} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}, & \psi_{y-} &= \frac{1}{\sqrt{2}} \begin{pmatrix} i \\ 1 \end{pmatrix}, \\ \psi_{z+} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix}, & \psi_{z-} &= \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{align} An advantage of using this convention is that the + and – wavefunctions may be related to one another, using the Pauli matrices themselves, by =i , = and = . Pauli vector The Pauli vector is defined by : \vec{\sigma} = \sigma_1 \hat{x} + \sigma_2 \hat{y} + \sigma_3 \hat{z} and provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows, : \begin{align} \vec{a} \cdot \vec{\sigma} &= \left(a_i \hat{x}_i\right) \cdot \left(\sigma_j \hat{x}_j\right) = a_i \sigma_j \hat{x}_i \cdot \hat{x}_j \\ &= a_i \sigma_j \delta_{ij} = a_i \sigma_i = \begin{pmatrix} a_3 & a_1 - ia_2 \\ a_1 + ia_2 & -a_3 \end{pmatrix} \end{align} using the . Further, : \det \vec{a} \cdot \vec{\sigma} = -\vec{a} \cdot \vec{a} = -\left|\vec{a}\right|^2, its eigenvalues being \pm |\vec{a}| , and moreover (see completeness, below) : \frac{1}{2} \operatorname{tr} \left(\left(\vec{a} \cdot \vec{\sigma}\right) \vec{\sigma}\right) = \vec{a} ~. Its (unnormalized) eigenvectors are : \psi_+ = \begin{pmatrix} a_3 + |\vec{a}| \\ a_1 + ia_2 \end{pmatrix}; \qquad \psi_- = \begin{pmatrix} ia_2 - a_1 \\ a_3 + |\vec{a}| \end{pmatrix}. Commutation relations The Pauli matrices obey the following relations: : \sigma_b = 2 i \varepsilon_{a b c}\,\sigma_c \, , and relations: : \{\sigma_a, \sigma_b\} = 2 \delta_{a b}\,I. where the is the , Einstein summation notation is used, is the , and is the identity matrix. For example, : \begin{align} \text{commutators} & & \text{anticommutators} & \, \\ \left\sigma_2\right &= 2i\sigma_3 \, & \left\{\sigma_1, \sigma_1\right\} &= 2I\, \\ \left\sigma_3\right &= 2i\sigma_1 \, & \left\{\sigma_1, \sigma_2\right\} &= 0\, \\ \left\sigma_1\right &= 2i\sigma_2 \, & \left\{\sigma_1, \sigma_3\right\} &= 0\, \\ \left\sigma_1\right &= 0 \, & \left\{\sigma_2, \sigma_2\right\} &= 2I\,. \\ \end{align} Relation to dot and cross product Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives : \begin{align} \left\sigma_b\right + \{\sigma_a, \sigma_b\} &= (\sigma_a \sigma_b - \sigma_b \sigma_a ) + (\sigma_a \sigma_b + \sigma_b \sigma_a) \\ 2i\varepsilon_{a b c}\,\sigma_c + 2 \delta_{a b}I &= 2\sigma_a \sigma_b \end{align} so that, each side of the equation with components of two -vectors and (which commute with the Pauli matrices, i.e., σqap)}} for each matrix and vector component (and likewise with ), and relabeling indices , to prevent notational conflicts, yields : \begin{align} a_p b_q \sigma_p \sigma_q & = a_p b_q \left(i\varepsilon_{pqr}\,\sigma_r + \delta_{pq}I\right) \\ a_p \sigma_p b_q \sigma_q & = i\varepsilon_{pqr}\,a_p b_q \sigma_r + a_p b_q \delta_{pq}I ~. \end{align} Finally, translating the index notation for the and results in | }} If i is identified with the pseudoscalar \sigma_x \sigma_y \sigma_z then the right hand side becomes a \cdot b + a \wedge b which is also the definition for the product of two vectors in geometric algebra. Some trace relations Following traces can be derived using the commutation and anticommutation relations. : \begin{align} \operatorname{tr}\left(\sigma_a\right) &= 0 \\ \operatorname{tr}\left(\sigma_a \sigma_b\right) &= 2\delta_{ab} \\ \operatorname{tr}\left(\sigma_a \sigma_b \sigma_c\right) &= 2i\varepsilon_{abc} \\ \operatorname{tr}\left(\sigma_a \sigma_b \sigma_c \sigma_d\right) &= 2\left(\delta_{ab}\delta_{cd} - \delta_{ac}\delta_{bd} + \delta_{ad}\delta_{bc}\right) \end{align} Exponential of a Pauli vector For : \vec{a} = a\hat{n}, \quad |\hat{n}| = 1, one has, for even powers, 2p, \ \ p = 0, 1, 2, 3, \ldots : (\hat{n} \cdot \vec{\sigma})^{2p} = I which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention. For odd powers, 2q + 1,\ \ q = 0, 1, 2, 3, \ldots : \left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} \, . , and using the , : \begin{align} e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} &= \sum_{k=0}^\infty{\frac{i^k \left\left(\hat{n} \cdot \vec{\sigma}\right)\right^k}{k!}} \\ &= \sum_{p=0}^\infty{\frac{(-1)^p (a\hat{n}\cdot \vec{\sigma})^{2p}}{(2p)!}} + i\sum_{q=0}^\infty{\frac{(-1)^q (a\hat{n}\cdot \vec{\sigma})^{2q + 1}}{(2q + 1)!}} \\ &= I\sum_{p=0}^\infty{\frac{(-1)^p a^{2p}}{(2p)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{q=0}^\infty{\frac{(-1)^q a^{2q+1}}{(2q + 1)!}}\\ \end{align} . In the last line, the first sum is the cosine, while the second sum is the sine; so, finally, | }} which is to , extended to . Note that : \deta(\hat{n} \cdot \vec{\sigma}) = a^2 , while the determinant of the exponential itself is just , which makes it the '''generic group element of '. A more abstract version of formula for a general matrix can be found in the article on . A general version of for an analytic (at a'' and −''a) function is provided by application of , : f(a(\hat{n} \cdot \vec{\sigma})) = I\frac{f(a) + f(-a)}{2} + \hat{n} \cdot \vec{\sigma} \frac{f(a) - f(-a)}{2} ~. The group composition law of A straightforward application of formula provides a parameterization of the composition law of the group . One may directly solve for in : \begin{align} e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} e^{i b\left(\hat{m} \cdot \vec{\sigma}\right)} &= I(\cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b) + i(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b )\cdot \vec{\sigma } \\ &= I\cos{c} + i (\hat{k} \cdot \vec{\sigma}) \sin{c} \\ &= e^{i c \left(\hat{k} \cdot \vec{\sigma}\right)}, \end{align} which specifies the generic group multiplication, where, manifestly, : \cos c = \cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b~, the . Given , then, : \hat{k} = \frac{1}{\sin c}\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} \sin a \sin b\right) ~. Consequently, the composite rotation parameters in this group element (a closed form of the respective in this case) simply amount to : e^{ic \hat{k}\cdot \vec{\sigma}}= \exp \left( i\frac{c}{\sin c} (\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b )\cdot \vec{\sigma}\right ) ~. (Of course, when is parallel to , so is , and a + b}}.) Adjoint action It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle , : e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma}~ e^{-i a\left(\hat{n} \cdot \vec{\sigma}\right)} = \vec{\sigma} \cos (2a) + \hat{n} \times \vec{\sigma} ~\sin (2a)+ \hat{n} ~ \hat{n} \cdot \vec{\sigma} ~ (1 - \cos (2a))~ . Completeness relation An alternative notation that is commonly used for the Pauli matrices is to write the vector index in the superscript, and the matrix indices as subscripts, so that the element in row and column of the -th Pauli matrix is . In this notation, the completeness relation for the Pauli matrices can be written : \vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{i=1}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}. : Proof: The fact that the Pauli matrices, along with the identity matrix I'', form an orthogonal basis for the complex Hilbert space of all 2 × 2 matrices means that we can express any matrix ''M as :: M = c I + \sum_i a_i \sigma^i : where c'' is a complex number, and ''a is a 3-component complex vector. It is straightforward to show, using the properties listed above, that :: \operatorname{tr}\, \sigma^i\sigma^j = 2\delta_{ij} : where "tr" denotes the , and hence that :: \begin{align} c &= \frac{1}{2}\operatorname{tr}\,M,\ \ a_i = \frac{1}{2}\operatorname{tr}\,\sigma^i M ~. \\3pt \therefore ~~ 2M &= I \operatorname{tr}\, M + \sum_i \sigma^i \operatorname{tr}\, \sigma^i M ~, \end{align} : which can be rewritten in terms of matrix indices as :: 2M_{\alpha\beta} = \delta_{\alpha\beta} M_{\gamma\gamma} + \sum_i \sigma^i_{\alpha\beta} \sigma^i_{\gamma\delta} M_{\delta\gamma}~, : where over the repeated indices γ'' and ''δ. Since this is true for any choice of the matrix M'', the completeness relation follows as stated above. As noted above, it is common to denote the 2 × 2 unit matrix by ''σ''0, so ''σ''0αβ'' = δ''αβ''. The completeness relation can alternatively be expressed as : \sum_{i=0}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma}~. The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the representation of 2 × 2 s' density matrix, (2 × 2 positive semidefinite matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of } as above, and then imposing the positive-semidefinite and conditions. For a pure state, in polar coordinates, \vec a = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta ) , the idempotent density matrix : \frac{1}{2}(1\!\! 1 + \vec a \cdot \vec \sigma) = \begin{pmatrix} \cos^2 \theta/2 & \sin \theta/2 ~ \cos\theta/2 ~e^{-i\phi} \\ \sin \theta/2 ~ \cos\theta/2 ~e^{i\phi} & \sin^2 \theta/2 \end{pmatrix} acts on the state eigenvector (\cos \theta/2 , e^{i\phi} \sin \theta/2) with eigenvalue 1, so like a for it. Relation with the permutation operator Let be the (also known as a permutation) between two spins and living in the space , : P_{ij}|\sigma_i \sigma_j\rangle = |\sigma_j \sigma_i\rangle \,. This operator can also be written more explicitly as , : P_{ij} = \frac{1}{2}(\vec{\sigma}_i\cdot\vec{\sigma}_j + 1)\,. Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates. SU(2) The group is the of matrices with unit determinant; its is the set of all anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the \mathfrak{su}_2 is the 3-dimensional real algebra by the set }. In compact notation, : \mathfrak{su}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2 , i \sigma_3 \}. As a result, each can be seen as an of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper }}, as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is , so that : \mathfrak{su}(2) = \operatorname{span} \left\{\frac{i \sigma_1}{2}, \frac{i \sigma_2}{2}, \frac{i \sigma_3}{2} \right\}. As SU(2) is a compact group, its is trivial. SO(3) The Lie algebra is to the Lie algebra , which corresponds to the Lie group , the of s in three-dimensional space. In other words, one can say that the are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though and are isomorphic as Lie algebras, and are not isomorphic as Lie groups. is actually a of , meaning that there is a two-to-one group homomorphism from to , see . Quaternions The real linear span of } is isomorphic to the real algebra of . The isomorphism from to this set is given by the following map (notice the reversed signs for the Pauli matrices): : 1 \mapsto I, \quad \mathbf{i} \mapsto - \sigma_2\sigma_3 = - i \sigma_1, \quad \mathbf{j} \mapsto - \sigma_3\sigma_1 = - i \sigma_2, \quad \mathbf{k} \mapsto - \sigma_1\sigma_2 = - i \sigma_3. Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order, : 1 \mapsto I, \quad \mathbf{i} \mapsto i \sigma_3, \quad \mathbf{j} \mapsto i \sigma_2, \quad \mathbf{k} \mapsto i \sigma_1. As the set of s ''U ⊂ ℍ forms a group isomorphic to , U'' gives yet another way of describing . The two-to-one homomorphism from to may be given in terms of the Pauli matrices in this formulation. Quaternions form a —every non-zero element has an inverse—whereas Pauli matrices do not. Physics Classical mechanics In , Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix ''P corresponding to the position \vec{x} of a point in space is defined in terms of the above Pauli vector matrix, : P = \vec{x} \cdot \vec{\sigma} = x\sigma_x + y\sigma_y + z\sigma_z ~. Consequently, the transformation matrix Q_\theta for rotations about the x''-axis through an angle ''θ may be written in terms of Pauli matrices and the unit matrix as : Q_\theta = 1\!\!1 \cos\frac{\theta}{2} + i\sigma_x \sin\frac{\theta}{2} ~. Similar expressions follow for general Pauli vector rotations as detailed above. References Category:Advanced mathematics